The Indian ace next faces 28th seed Sorana Cirstea of Romania, who beat countrywoman Edina Gallovits 7-5, 6-1.
Cirstea had reached the quarter-finals of the French Open, accounting for, amongst others, second seeded Serbian Jelena Jankovic.
This was 78-ranked Sania's third victory over Groenefeld, having scored over her in their meetings in 2004 and 2005.
After taking the first set comfortably Sania faced stiff resistance in the second. She had three breakpoints in the first game but the German saved all before breaking in the next and then holding for a 3-0 lead.
Sania immediately returned the break, but dropped serve again in the sixth game to squander an opportunity to draw abreast. The Indian then saved two set points in the eighth game but unforced errors handed the German the set.
A keen contest was expected in the decider, but it was the Indian who returned to the court firing on all cylinders as she broke in the first game itself and crusied through the next four to establish a 4-1 lead.
The next three games went with serve and the Indian wrapped up the set and match comfortably.
Other first round results (prefix number denotes seeding):
24-Maria Sharapova (Russia) beat Viktoriya Kutuzova (Ukraine) 7-5, 6-4
16-Zheng Jie (China) beat Kristina Barrois (Germany) 7-6(2), 7-6(4)
Regina Kulikova (Russia) beat Karolina Sprem (Croatia) 4-6, 7-5, 6-3
Daniela Hantuchova (Slovakia) beat Laura Robson (Britain) 3-6, 6-4, 6-2
28-Sorana Cirstea (Romania) beat Edina Gallovits (Romania) 7-5, 6-1
27-Alisa Kleybanova (Russia) beat Sesil Karatantcheva (Kazakhstan) 6-2, 7-5
Gisela Dulko (Argentina) beat Stephanie Foretz (France) 6-3, 7-5
31-Anastasia Pavlyuchenkova (Russia) beat Petra Cetkovska (Czech Republic) 6-2, 6-2.